How can I create and keep these faces together nicely?
My previous answer won't give a correct method.

Actually the Snap option would work correctly if it was compatible with GG (slide vertex along an edge) but it's not the case yet.

What you should do to get edges perpendicular to the X, Y and Z axis is use the knife tool: K to activate, C to cut orthogonally to an axis, Z to cut through.



Once you've created an edge, you can cut again to create a parallel one, or simply select it and bevel with CtrlB.



Then again use the knife tool for vertical edges:



And when you begin to extrude, constraint on the axis you want: E to extrude then X to move the extrusion on the X axis.